Problem 16 - Advanced Engineering Mathematics

Question

If \(\mathbf{A}=\left(\begin{array}{rr}-2 & 0 \\ 4 & 1 \\ 7 & 3\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}3 & -1 \\ 0 & 2 \\\ -4 & -2\end{array}\right)\), find (a) \(\mathbf{A}-\mathbf{B}\) (b) \(\mathbf{B}-\mathbf{A}\), (c) \(2(\mathbf{A}+\mathbf{B})\)

Step-by-Step Solution

Answer

(a) \(\mathbf{A} - \mathbf{B} = \begin{pmatrix}-5 & 1 \\ 4 & -1 \\ 11 & 5\end{pmatrix}\), (b) \(\mathbf{B} - \mathbf{A} = \begin{pmatrix}5 & -1 \\ -4 & 1 \\ -11 & -5\end{pmatrix}\), (c) \(2(\mathbf{A} + \mathbf{B}) = \begin{pmatrix}2 & -2 \\ 8 & 6 \\ 6 & 2\end{pmatrix}\).

Step 1: Set Up the Matrices for Subtraction A-B

To perform the operation \(\mathbf{A} - \mathbf{B}\), align the elements of matrices \(\mathbf{A}\) and \(\mathbf{B}\). We will subtract each corresponding element of \(\mathbf{B}\) from \(\mathbf{A}\).Matrix \(\mathbf{A}\): \(\begin{pmatrix}-2 & 0 \ 4 & 1 \ 7 & 3\end{pmatrix}\)Matrix \(\mathbf{B}\): \(\begin{pmatrix}3 & -1 \ 0 & 2 \ -4 & -2\end{pmatrix}\)

Step 2: Perform the Subtraction A-B

Subtract each element of \(\mathbf{B}\) from the corresponding element in \(\mathbf{A}\):\[\mathbf{A} - \mathbf{B} = \begin{pmatrix}-2 - 3 & 0 - (-1) \ 4 - 0 & 1 - 2 \ 7 - (-4) & 3 - (-2)\end{pmatrix} = \begin{pmatrix}-5 & 1 \ 4 & -1 \ 11 & 5\end{pmatrix}\]

Step 3: Set Up the Matrices for Subtraction B-A

To find \(\mathbf{B} - \mathbf{A}\), we reverse the subtraction order and subtract each element of \(\mathbf{A}\) from \(\mathbf{B}\) corresponding elements.Matrix \(\mathbf{A}\): \(\begin{pmatrix}-2 & 0 \ 4 & 1 \ 7 & 3\end{pmatrix}\)Matrix \(\mathbf{B}\): \(\begin{pmatrix}3 & -1 \ 0 & 2 \ -4 & -2\end{pmatrix}\)

Step 4: Perform the Subtraction B-A

Subtract each element of \(\mathbf{A}\) from the corresponding element in \(\mathbf{B}\):\[\mathbf{B} - \mathbf{A} = \begin{pmatrix}3 - (-2) & -1 - 0 \ 0 - 4 & 2 - 1 \ -4 - 7 & -2 - 3\end{pmatrix} = \begin{pmatrix}5 & -1 \ -4 & 1 \ -11 & -5\end{pmatrix}\]

Step 5: Set Up the Matrices for Addition A+B

Add matrices \(\mathbf{A}\) and \(\mathbf{B}\) by aligning and adding each corresponding element.Matrix \(\mathbf{A}\): \(\begin{pmatrix}-2 & 0 \ 4 & 1 \ 7 & 3\end{pmatrix}\)Matrix \(\mathbf{B}\): \(\begin{pmatrix}3 & -1 \ 0 & 2 \ -4 & -2\end{pmatrix}\)

Step 6: Perform the Addition A+B

Add each element of \(\mathbf{A}\) to the corresponding element of \(\mathbf{B}\):\[\mathbf{A} + \mathbf{B} = \begin{pmatrix}-2 + 3 & 0 + (-1) \ 4 + 0 & 1 + 2 \ 7 + (-4) & 3 + (-2)\end{pmatrix} = \begin{pmatrix}1 & -1 \ 4 & 3 \ 3 & 1\end{pmatrix}\]

Step 7: Multiply the Sum by 2

Multiply each element of the resulting matrix from \(\mathbf{A} + \mathbf{B} \) by 2:\[2(\mathbf{A} + \mathbf{B}) = 2 \begin{pmatrix}1 & -1 \ 4 & 3 \ 3 & 1\end{pmatrix} = \begin{pmatrix}2 & -2 \ 8 & 6 \ 6 & 2\end{pmatrix}\]

Key Concepts

Matrix Subtraction

Matrix subtraction is a fundamental operation where you subtract one matrix from another. To perform this operation, the matrices must be of the same dimension. This means they must have the same number of rows and columns.
Matrix subtraction involves aligning both matrices and subtracting each element of the second matrix from the corresponding element in the first matrix. For example, if you have two matrices, \[\mathbf{A} = \begin{pmatrix} a & b \ c & d \end{pmatrix} \quad \text{and} \quad \mathbf{B} = \begin{pmatrix} w & x \ y & z \end{pmatrix},\]the subtraction \( \mathbf{A} - \mathbf{B} \) is calculated as follows:\[\mathbf{A} - \mathbf{B} = \begin{pmatrix} a-w & b-x \ c-y & d-z \end{pmatrix}.\]When performing such operations:

• Ensure both matrices are of identical dimensions.
• Treat each element individually.
• Apply subtraction element-wise.

It's essential to carefully align elements since matrix subtraction is not commutative; \( \mathbf{A} - \mathbf{B} eq \mathbf{B} - \mathbf{A} \) in general.

Matrix Addition

Matrix addition is another key operation in matrix algebra, similar in its simplicity to matrix subtraction. Just like with subtraction, the matrices involved must have the same dimensions. This means the matrices should have the same number of rows and columns.
To add two matrices, simply align them and add their corresponding elements together. For instance, if you have matrices:\[\mathbf{A} = \begin{pmatrix} a & b \ c & d \end{pmatrix} \quad \text{and} \quad \mathbf{B} = \begin{pmatrix} w & x \ y & z \end{pmatrix},\]The addition, \( \mathbf{A} + \mathbf{B} \), is computed as:\[\mathbf{A} + \mathbf{B} = \begin{pmatrix} a+w & b+x \ c+y & d+z \end{pmatrix}.\]Keep in mind:

• Matrix addition is commutative: \( \mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A} \).
• The operation is straightforward and involves adding elements in the same position across matrices.
• The resultant matrix retains the same dimensions.

Matrix addition can be used in various applications, from solving linear equations to modeling transformations in computer graphics.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a constant, known as a scalar. This simple operation can modify a matrix by scaling its size, but it does not change its shape.
Let's consider a matrix \( \mathbf{A} \) and a scalar \( k \):\[\mathbf{A} = \begin{pmatrix} a & b \ c & d \end{pmatrix}, \quad \text{and let the scalar be } k.\]The scalar multiplication \( k \mathbf{A} \) results in:\[k \mathbf{A} = \begin{pmatrix} ka & kb \ kc & kd \end{pmatrix}.\]Here's what you'll want to keep in mind:

• Scalar multiplication affects each matrix element equally.
• It is distributive; when \( (a+b) \mathbf{A} \) is considered, it can be rewritten as \( a \mathbf{A} + b \mathbf{A} \).
• Preserves the matrix's dimensions.

This operation is particularly useful when you're looking to scale a matrix, especially in applications such as digital imaging and physics simulations, where adjustments to entire matrices are needed.